Reinforcing Old Knowledge

This semester at school I am taking Functional Analysis.  In Functional analysis, one studies functionals on spaces of functions, as in Real Analysis one studies functions on the space of real numbers.  The first class was an opportunity to see some material with which I was already familiar and to reinforce concepts I had learned in a previous class.

Our first topic of study was metric spaces.  A metric generalizes the concept of distance between two elements in a space.  Pairs of us students were given a space, and we were to propose a metric for it.  Our given space was the space of step functions (each element of the space is a step function); our idea was to partition the space into intervals where each step function has a constant value.  Then we can find the area of the rectangle between the functions in each interval.  Finally, we can add the areas together to get our proposed metric.  This seemed correct, but I needed a written description in mathematical language of what we had found to convince me that it was indeed correct.

I realized that I knew how to write a step function: \displaystyle \sum^{\infty}_{i=1} a_i \chi_{E_i}.  Writing a second step function in this notation, but using F as the set variable and j as the index, and using the concept of measure (\mu), our metric can be written as follows: \displaystyle \sum^{\infty}_{i=1} |a_i - b_j| \mu_{E_i \cap F_j}.  Voila!

One pair of students was given the space of sequences with the condition that for each sequence, the sum of the squares of the terms converged.  A sequence in this space is \{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \} where 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + ... = \frac{\pi^2}{6}.  This space is called l_2.  This was familiar to me as I had studied L_2, the space of Lebesgue integrable functions with 2-norm, in a previous class.  It turns out that l_2 and L_2 are closely related: a relationship that we will explore later in the class.

This has been my journey in mathematics: to learn new concepts and connect them to old concepts.  The opportunity to recall step functions, the concept of measure, and the spaces of Lebesgue integrable functions and to apply these concepts to solving a new, albeit contrived, problem made for a fun first class.

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1 Comment

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One response to “Reinforcing Old Knowledge

  1. I was surprised in the good way in taking functional analysis just from how much more naturally everything seemed to flow together compared to real analysis. That might reflect just that the analysis way of thinking about things is easier after a couple terms of real analysis, but I do suspect there’s something about functional that’s just easier.

    I also keep getting surprised by how often characteristic functions, for all that they seem too simple to deal with, make hard problems easy.

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